Fenchel conjugate [1]} (a.k.a. convex conjugate) is the key tool to transform the original problem into bi-level optimization problem. For the general convex function \(h(\cdot )\) , its conjugate function is defined as
\(h^{*}(\mathbf {U})\stackrel{\triangle }{=}\sup \limits _{\mathbf {X}}\langle \mathbf {U}, \mathbf {X}\rangle -h(\mathbf {X}). \nonumber \) 
